split object into equal halves
'Knife project' works well here :


Add a plane
Place one of its edges along the middle (at the exact center)
You may work in a ortho view and along the axis (here top ortho)
Stretch it so that it covers your mesh
Select your mesh (keeping the plane selected) and enter edit mode
Hit 'knife project' and make sure the option is 'cut through'
Select the half to remove and add the mirror modifier